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Full text of "Mathematical And Physical Papers - Iii"

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In the next place, the formula (13) applies to motion in one dimension only. Eat had we employed the general equations (1), (2), which relate to motion in three dimensions, we should have obtained the same partial differential equation as (7), with the exception that the last term outside the brackets would have been replaced by
If now we take the case next in order of simplicity, in which the motion is symmetrical a/bout a centre, and put r for the distance of any point from the centre, we shall get for the determination of rs the same partial differential equation as (7), with the exception that x will "be xeplaced by r. To obtain, therefore, the integral corresponding to (1 3), it will be sufficient to replace x by r and divide the second 3nenaber by r. This integral would apply to the case of the disturbance produced by a vibrating spherical body, in which the motion, is supposed to be symmetrical with respect to the centre. A. nd in the more general case of a vibrating body of irregular form, or a musical instrument, or any other source of sound, the conclusions would doubtless be the same as to their leading features.
There remains a more important point to be considered before we apply the formula (13) to the vibrations of air within a long tube. At first sight it might seem that the radiation of heat within a tube must take place in a manner altogether different from that in which it would take place in free air. But a little consideration will show, I think, that such is not the case. Of the heat radiating from any particle of air which has been slightly heated by condensation, any particular ray is incident on the side of the tube, where it is partly absorbed, partly reflected, and, it may ho, partly scattered. The reflected ray, or any one of the scattered rays, is again incident on the side of the tube, where a good portion is absorbed, and so on. The small quantity of radiant heat which remains after three or four reflexions may be regarded as insensible. Now since radiant heat travels with a velocity equal to, or at a,riy rate comparable with, that of light, we may neglect as altogether insensible the time which any portion of heat, once become t-adiant, takes to be absorbed. Moreover, we may neglect the small portion of heat reabsorbed by the air